Notes on character sums and complex functions over nite elds N. V. Proskurin Abstract. The intent of this notes is to present briey the complex functions over nite elds theory. That includes: (a) additive and multiplicative characters; (b) Gauss and Jacobi sums, other trigonometric sums; (c) Fourier expansion, power series expansion, dierentiation; (d) special functions like Hermit polynomials, hypergeometric functions and so on. The theory has been developed through parallels with the classical functions theory. The basis for this parallel is the analogy between Gauss sums and the gamma function. 1. Preliminaries The notation we use is very standard, and we only summarize here the most common. Given prime p, let F be the nite eld with = p l elements and with prime subeld F p = Z/pZ. Let trace: F F p be the trace function. We write F for the multiplicative group of F. Fix (once for all) a non-trivial additive character e : F C. With some h F, one has e (x) = exp ( 2πi trace(hx)/p ) for all x F. We write F for the group of multiplicative characters of F, i. e. for the group of homomorphisms : F C. Extend each multiplicative character to all of F by setting () =. We write ɛ for the trivial character, ɛ(x) = 1 for all x F, ɛ() =. Dene δ : F C by setting δ() = 1 and δ(x) = for all x F. 2. Background The Gauss sum G() = (x)e (x) with F, being considered as complex x F function on F, is uite similar to the Euler gamma function Γ (s) = exp( x) x s 1 dx for s C with Re s >.
2 N. V. Proskurin For better understanding, note that x exp( x) is a character of the additive group of real numbers eld, that x x s is a character of the multiplicative group of positive real numbers, and that the integration over dx/x is invariant under multiplicative translations. Similarly, we see that the Jacobi sum J(α, β) = x F α(x)β(1 x) with α, β F is the analogue of the Euler beta function B(a, b) = 1 x a 1 (1 x) b 1 dx for a, b C, Re a >, Re b >. It occurs that a lot of identities satised by the gamma and beta functions have nite eld analogues. Say, the formulas G() G() = ( 1) and J(α, β) = G(α)G(β), G(αβ) where ɛ and αβ ɛ, are the nite eld analogues of the reection formula Γ (s) Γ (1 s) = π for s C sin πs and Γ (a)γ (b) B(a, b) = with a, b C. Γ (a + b) The analogy observed dates back to the nineteenth century. For more advanced analogy we may look on the Gauss multiplication formula for the gamma function, m 1 n= ( Γ s + n ) = (2π) (m 1)/2 m 1/2 ms Γ (ms) m with any s C and integer m 1. Given a nite eld F, a character ψ F and an integer m ( 1), we have a nite eld counterpart of the multiplication formula. That is the DavenportHasse [1 relation { G(ψ) = ψ(m) } m G() G(ψ m ), where the products range over F under the assumption m = ɛ. 3. On cubic exponential and Kloosterman sums A very instructive sample of analogy between classical special functions and character sums is given by Iwaniec and Duke [2. Consider the integral representation K 1/3 (u) = u 1/3 ) exp ( t u2 dt 4t (2t) 4/3
Functions over nite elds 3 for the BesselMacdonald function and the Nicholson formula for the Airy integral cos(t 3 + tv) dt = v1/2 3 K 1/3(2(v/3) 3/2 ), which are valid at least for real positive u, v. From these two formulas, it follows exp ( i (cx 3 + x) ) dx = 3 1/2 ( x c ) 1/3 exp ( x 1 27cx ) dx x with any real c >. Assume 3 ( 1). This case there exists cubic character ψ of F and cubic Kloosterman sums, which one can treat as analogue of the integral in the right-hand side. Following the analogue, Iwaniec and Duke deduced very important formula which relates cubic Kloosterman sum to cubic exponential sum. That is e (cx 3 + x) = ψ(xc 1 ) e (x (27cx) 1 ) for c F. x F x F The proof in [2 involves DavenportHasse relation and Fourier transform. For extended result see [3. 4. Elements of analysis Consider the complex vector space Ω of all functions F C supplied with the inner product f, g = x F f(x)g(x) for all f, g Ω. The additive characters form an orthogonal bases for the Ω. Given a function F : F C, its additive Fourier transform is the function ˆF : F C dened by ˆF (x) = y F F (y)e (yx) for all x F. The Fourier inversion formula F (z) = 1 ˆF (x)e ( xz) for all z F x F allows one to recover F from ˆF and can be considered as the expansion of F over basis consisting of additive characters. Given a function F : F C, its multiplicative Fourier transform is the function F : F C dened by F () = x F F (x)(x) for all F.
4 N. V. Proskurin The Fourier inversion formula F (z) = 1 1 F F ()(z) for all z F allows one to recover F from F. Given a function F : F C, one has a similar expansion with one additional term F (z) = F () δ(z) + F C (z) with C = 1 1 F (). One can treat the sum over as the sum over ɛ, ρ, ρ 2,... ρ 1, whenever ρ generates the group F. So, that is a nite eld analogue of power series expansion. The multiplicative characters together with δ form an orthogonal bases for the Ω. For example, given ρ F ρ(1 + x) = δ(x) + 1 k= and x F, one has the expansion ρ ρ (x) with F = ( 1) J(ρ, ), which is nite eld analogue of the classical binomial formula r r r (1 + x) r = x k r! with = k k (r k)! k!, x C. For another example, let F be additive character e. One has e ( z) = 1 + 1 F (z) G() for all z F. That is a nite eld analogue for the power series expansion for the exponent. We refer to Greene [4 for both examples and for properties of the binomial coecients. 5. Dierentiation Given character F, dene the linear operator D : Ω Ω by D F (x) = 1 G() for all F : F C and x F. We nd easily t F F (t) (x t) D ɛ F (x) = F (x) t F F (t), 1 G() D F (x) = 1 t F F (t) (t x)
Functions over nite elds 5 for all F and x as above and ɛ. According to Evans [5, D F is the derivative of order of F. This denition is motivated by the Cauchy integral formula 1 n! f (n) (x) = 1 f(t) dt 2πi (t x) n+1 for the derivative f (n) of any order n of the function f. One nds easily some standard properties. Say, D takes constant functions to zero function, whenever ɛ. Then, D α D β = D αβ for characters α, β subject to αβ ɛ. Also, given two functions E and F, x F, and the character ν we have the formula for integration by parts E(x) D ν F (x) = ν( 1) F (x) D ν E(x) x F x F and the Leibniz rule for the ν-th derivative of the product D ν EF (x) = 1 G(µ)G(µν) D µ E(x) D νµ F (x). 1 G(ν) µ F Given any character ν, let F (x) = e ( x) for all x F. This case we have D ν F = F. For any function F : F C and a F, we have expansion F (x) = 1 G(ν)D ν F (a) ν(a x) 1 ν F for all x F, x a. That is a nite eld analogue of the Taylor expansion. 6. Hermite character sums Given any character ν F, let H ν (x) = 1 ν(u)e (u 2 + 2ux) G(ν) u F for all x F. The denition is given in [5 as nite eld analogue of the classical Hermite polynomials. We nd in [5 a lot of formulas involving the character sums H ν which are uite similar to that for the Hermite polynomials H n. Say, we have H n ( x) = ( 1) n H n (x) for all x R and integer n, and we have H ν ( x) = ν( 1) H ν (x) for all x F, ν F. Also, for all x F, one has H ν (x) = ν( 1)F (x) 1 D ν F (x), where F (x) = e (x 2 ). That is a nite eld analogue of the Rodriguez formula H n (x) = ( 1) n exp(x 2 ) dn dx n exp( x2 ), x R, for the classical Hermite polynomials H n. In a similar manner one can treat the Legendre polynomials, the Bessel functions and other classical polynomials and special functions.
6 N. V. Proskurin 7. Hypergeometric functions For the hypergeometric function 2 F 1 one has the Euler integral representation 2F 1 (a, b; c; x) = Γ(c) Γ(b) Γ(c b) 1 t b (1 t) c b (1 xt) a dt t (1 t). Greene [4 dened the hypergeometric functions on F by [ α, β 2F 1 x = ɛ(x) β( 1) β(t) β(1 t) α(1 xt) t F for any characters α, β, F and x F. With this denition and notation one has power series expansion [ α, β 2F 1 x = α β (x), 1 and one has the more general denition [ α, α 1,..., α n n+1f n x = α α1 αn... (x) β 1,..., β n 1 β 1 β n for any integer n 1 and characters α,... β n F. Like their classical counterparts, hypergeometric functions over nite elds satisfy many transformation identities. Say, as analogue for Pfa's transformation [6, [4 we have 2F 1 [ α, β F F [ α, β x = β(1 x) 2 F 1 x x 1 for any characters α, β, F and x F, x 1. Let us turn to the function 3 F 2. In the classical context, there are some summation formulas. We mean the formulas of Saalschütz, Dixon, Watson, Whipple. Greene [4 gives analogues for each of them. Say, as analogue for Saalschütz's formula [6 we have [ α, β, β 3F 2 1 = β( 1) 1 βρ ρ, αβρ αρ ρ βρ( 1). α As one more example, consider Clausen's famous classical identity [7 3F 2 (2c 2s 1, 2s, c 1/2; 2c 1, c; x) = 2 F 1 (c s 1/2, s; c; x) 2, which was utilized in de Branges' proof of the Bieberbach conjecture. By Evans and Greene [8, one has a nite eld analogue of this formula. That is [ α 3F 2 2, α 2, φ (x) φ(1 x) (4) J(α, α) [ αφ, α 2, 2 2 x = +, 2F 1 x J(α, α) where x F, x 1, φ is the uadratic character (φ ɛ, φ 2 = ɛ), and it is assumed that φ, α 2 ɛ,, 2.
Functions over nite elds 7 A study of special function analogues over nite elds serve as a useful tool when applied to problems related to character sums. For more transformation formulas, farther developments and applications we refer to [9, 1, 11, 12, 13, 14, 15. One can nd Magma code to compute the hypergeometric functions over nite elds in the dissertation [1. References [1 H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. reine angew. Math., 172, 1934, 151182. [2 W. Duke and H. Iwaniec, A relation between cubic exponential and Kloosterman sums, Contemporary Mathematics, 143, 1993, 255258. [3 Zhengyu Mao, Airy sums, Kloosterman sums, and Salié sums, Journal of Number Theory, 65, 1997, 31632. [4 J. Greene, Hypergeometric functions over nite elds, Transactions of the American Math. Soc., 31, No. 1, 7711, 1987. [5 R. J. Evans, Hermite character sums, Pacic Journal of Mathematics, 122, No. 2, 35739, 1986. [6 L. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, 1966. [7 W. N. Bailey, Generalized hypergeometric series, StrechertHafner, New York, 1964. [8 R. Evans, J. Greene, Clausen's theorem and hypergeometric functions over nite elds, Finite Fields and Their Applications 15, 29, 9719. [9 J. Greene, D. Stanton, A character sum evaluation and Gaussian hypergeometric series, Journal of Number Theory, 23, no. 1, 136148, 1986. [1 C. Lennon, Arithmetic and analytic properties of nite eld hypergeometric functions, Submitted for the degree of PhD at the Massachusetts Inst. of technology, June 26. [11 J. Rouse, Hypergeometric functions and elliptic curves, Ramanujan Journal, 12, no. 2, 19725, 26. [12 R. Barman, G. Kalita Elliptic curves and special values of Gaussian hypergeometric series, Journal of Number Theory, 133, issue 9, pages 3993111, September 213. [13 A. Deines, J. G. Fuselier, L. Long, H. Swisher, Fang-Ting Tu, Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions, Directions in Number Theory, Springer, Cham. Association for women in mathematics series, vol. 3, 216. [14 J. Fuselier, L. Long, R. Ramakrishna, H. Swisher, Fang-Ting Tu, Hypergeometric functions over nite elds, arxiv:151.2575v2 [math.nt, 1 April 216. [15 R. Evans, J. Greene, A uadratic hypergeometric 2F 1 transformation over nite elds, Proc. Amer. Math. Soc. 145, 217, 171176. N. V. Proskurin St. Petersburg Department of Steklov Institute of Mathematics RAS 19123, Fontanka 27, St. Petersburg, Russia e-mail: np@pdmi.ras.ru